Problem Sets: Continuity (连续性)

Last Update: 2026-09-29

Remark:

• In accordance with the course sequence (late-transcendentals), this problem set does not yet include problems involving exponential functions, inverse functions, or logarithms. These topics will be covered in subsequent problem sets.

• This problem set focuses on limits. Do not use L’Hôpital’s Rule or Taylor expansions; they are not needed here.

• The Additional and Advanced Exercises are somewhat more involved than the standard exercises. They may require multi-step reasoning, conceptual analysis, modest proof components, or a combination of several ideas.

Continuity

Basic Continuity

• Let $f(x)= \begin{cases}3+x^{2}, & x \leq 0, \\ \dfrac{\sin 3 x}{x}, & x>0 \end{cases}$. Is $f(x)$ continuous?

• Find the constants $a$ and $b$ for which the function $f(x)= \begin{cases}\dfrac{\cos\pi x}{x^{2}+ax+b}, & x \neq \dfrac{1}{2}, \\ 2, & x = \dfrac{1}{2} ,\end{cases}$ is continuous at $x = \dfrac{1}{2}$.

• What is the number of removable discontinuities of the function $f(x) = \dfrac{x - x^{3}}{\sin \pi x}$?

• What is the number of infinite discontinuities of the function $f(x) = \dfrac{x^{2} - x}{x^{2} - 1}\sqrt{1 + \dfrac{1}{x^{2}}} $?

• Prove that $f$ is continuous at $c$ if and only if $\displaystyle\lim_{h\to 0} f(c+h) = f( c )$.

• Prove that $f(x) = \sin x$ is continuous at every point.

• Prove that all six trigonometric functions are continuous wherever they are defined.

• Give an example of functions $f$ and $g$ such that $f(g(x))$ is continuous but $g$ has at least one discontinuity.

• Show that the $f(x)=\begin{cases}x^{2} \sin (1 / x), & x<0, \\ \sqrt{x}, & x\geq 0,\end{cases}$ is continuous on $(-\infty,\infty)$.

• The function $f(x) = \begin{cases}\dfrac{1 - \cos\sqrt{x}}{ax}, & x > 0, \\b, & x \leq 0 \end{cases}$ is continuous at $x = 0$. Find $a$ and $b$.

• Let the function $f(x) = \begin{cases} -1, & x < 0, \\ 1, & x \geq 0, \end{cases} $ and $g(x) = \begin{cases} 2 - ax, & x \leq -1, \\ x, & -1 < x < 0, \\ x - b, & x \geq 0. \end{cases} $. If $ f(x) + g(x) $ is continuous on $ \mathbb{R} $, then find $a$ and $b$.

• Let the function $f(x) = \begin{cases} x^{2} + 1, &

  x

  \leq c, \\ \dfrac{2}{

  x

  }, &

  x

  > c \end{cases}$ be continuous on $(-\infty, \infty) $. Find $c$.

The Intermediate Value Theorem

• Show that the equation $x^{3} - 15x -\cos x = 0$ has at least three solutions in the interval $[-4,4]$.

• Suppose $f(x)$ is continuous on $[0,1]$, and for any $x\in [0,1]$, we have $0 \leq f(x) \leq 1$. Prove that there exists a point $c \in [0,1]$, such that $f(c) = c$.
  • Give an example showing that if $f$ is continuous and $0 < f(x) < 1$ for $0 < x < 1$, then there does not need to be a $c$ in $(0,1)$ such that $f(c) = c$.
  • Give an example showing that if $0 \le f(x) \le 1$ for $0 \le x \le 1$, but $f$ is not necessarily continuous, then there does not need to be a $c$ in $[0,1]$ such that $f(c) = c$.

• Use the IVT to show that if $f$ is continuous and one-to-one on an interval $[a,b]$, then $f$ is either an increasing or a decreasing function.

Additional and Advanced Exercises

• Let $f(x)= \begin{cases}x^{\alpha}\sin\dfrac{1}{x^{\beta}}, & x \neq 0, \\ 0, & x=0 \end{cases}$. Find $\alpha>0$ and $\beta>0$ such that $f(x)$ is continuous at $x=0$.

• Let $f(x)$ be continuous on $(-\infty, \infty)$, and $f(f(x)) = x$. Prove that there exists at least one point $\xi \in (-\infty, \infty)$ such that $f(\xi) = \xi $.

• Continuous at Only One Point. Show that the following function is continuous only at $x = 0$: \(f(x)= \begin{cases} x & \text{for } x \text{ rational}\\ -x & \text{for } x \text{ irrational} \end{cases}\)

• Show that $f$ is a discontinuous function for all $x$, where $f(x)$ is defined as follows: \(f(x)= \begin{cases} 1 & \text{for } x \text{ rational}\\ -1 & \text{for } x \text{ irrational} \end{cases}\) Show that $f^{2}$ is continuous for all $x$.

• Here is a function with strange continuity properties: \(f(x) = \begin{cases} \frac{1}{q}, & x = p/q \text{ is the rational and written in lowest terms}, \\\\ 0, & x \text{ is an irrational number}= 0, \end{cases}\)

  1. Show that $f$ is discontinuous at $c$ if $c$ is rational. Hint: There exist irrational numbers arbitrarily close to $$c$.
  2. Show that $f$ is continuous at $c$ if $c$ is irrational. Hint: Let $I$ be the interval ${x : \lvert x - c\rvert < 1}$. Show that for any $Q > 0$, I contains at most finitely many fractions $p/q$ with $q < Q$. Conclude that there is a $\delta$ such that all fractions in ${x : \lvert x - c\rvert < \delta}$ have a denominator larger than $Q$.